Licensing Uncertain Patents: Per-Unit Royalty vs

Up-Front Fee

David Encaoua, Yassine Lefouili

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David Encaoua, Yassine Lefouili. Licensing Uncertain Patents: Per-Unit Royalty vs Up-FrontFee. 3rd European Conference on Competition and Regulation, Jul 2008, Athenes, Greece.<hal-00318208>

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Licensing Uncertain Patents:

Per-Unit Royalty vs. Up-Front Fee�

David Encaouayand Yassine Lefouiliz

Paris School of Economics, University of Paris-I Panthéon Sorbonne

August 2008

Abstract

We examine the implications of uncertainty over patent validity on patenthold-

ers� licensing strategies. Two licensing schemes are investigated: the per-unit

royalty rate and the up-front fee. We provide conditions under which uncertain

patents are licensed in order to avoid patent litigation. It is shown that while it

is possible for the patentholder to reap some "extra pro�t" by selling an uncer-

tain patent under the per-unit royalty scheme, the opportunity to do so does not

exist under the up-front fee scheme. We also establish that the relatively high

bargaining power the licensor has even when its patent is weak can be reduced if

the patentholder cannot refuse to license an unsucessful challenger or if collective

challenges are allowed for. Furthermore we show that the patentee may prefer to

license through the per-unit royalty mechanism rather than the �xed fee mecha-

nism if its patent is weak whereas it would have preferred the latter to the former

if the patent were strong. This �nding gives a new explanation as to why the

per-unit royalty scheme may be preferred by a patentholder to the up-front fee

scheme.�We are grateful to Rabah Amir, Vincenzo Denicolo, David Ulph and Georg von Graevenitz for

useful comments and helpful discussions. We would also like to thank participants at the CRESSE2008 in Athens.

yAddress: Centre d�Economie de la Sorbonne, 106-112, Bd de l�Hôpital 75647 cedex 13, Paris,France. E-mail: encaoua@univ-paris1.fr.

zAddress: Centre d�Economie de la Sorbonne, 106-112, Bd de l�Hôpital 75647 cedex 13, Paris,France. E-mail: yassine.lefouili@m4x.org.

1

1 Introduction

Licensing intellectual property is a key element in the innovation process and its dif-

fusion. A license is a contract by which the owner of intellectual property authorizes

another party to use it, in exchange for payment.1 The properties and virtues of li-

censing (Kamien, 1992, Scotchmer, 2004) have mainly been analyzed in a framework in

which intellectual property rights guarantee perfect protection and give their owners a

right to exclude as strong as physical property rights do. This framework does not cor-

respond to what we observe in practice. In the real world patents do not give the right

to exclude but rather a more limited right to "try to exclude" by asserting the patent

in court (Ayres and Klemperer, 1999, Shapiro, 2003, Lemley and Shapiro, 2005). The

exclusive right of a patentholder can be enforced only if the court upholds the patent�s

validity. For this reason, patents are considered as probabilistic rights rather than iron-

clad rights. This paper is devoted to the analysis of licensing patents that are uncertain,

i.e. patents that have a positive probability to be invalidated by a court if they are

challenged.2

Many reasons explain the inherent uncertainty attached to a patent. First, the

standard patentability requirements, namely the subject matter, utility, novelty and

non-obviousness (or inventive step in Europe) are di¢ cult to assess by patent o¢ ce

examiners. Legal uncertainty over the patentability standards is especially pervasive in

the new patenting subject matters for which the prior art is rather scarce, like software

or business methods. Moreover, the claims granted by the patent o¢ ce are supposed

to delineate the patent scope, but their ex post validation depends on the judicial

doctrine adopted by the court, and it may be di¢ cult for a patentholder and a potential

infringer to know exactly what the patent protects. Second, the resources devoted to

the patentability standards review by the patent o¢ ce are in general insu¢ cient to

allow an adequate review of each patent application.3 Many innovations are granted

1According to some surveys (Taylor and Silberstone,1973, Rostoker,1984, and Anand and Khanna,2000), the per-unit royalty rate and the �xed fee mechanism are the most frequent licensing schemes.

2Uncertainty does not necessarily imply asymmetric information or di¤erent beliefs about patent�svalidity among involved parties. Uncertainty may occur even if the parties share the same beliefs onthe patent validity. For a di¤erent view, see Bebchuk (1984), Reinganum and Wilde (1986), Meurer(1989), Hylton (2002).

3The average time spent by an examiner on each patent is about 15-20 hours in the USPTO (Ja¤e& Lerner, 2004) and around 30 hours in the EPO. The gap between the massive growth of patentapplications and the insu¢ cient resources at the patent o¢ ce creates a "vicious circle" (Caillaud andDuchêne, 2005). Incentives to �le "bad applications" increase the patent o¢ ce overload, and a larger

2

patent protection even though they do not meet patentability standards. This results

in many "weak patents", i.e. patents that have a high probability to be invalidated by

a court if they are challenged. Finally, it has been argued that incentives inside the

patent o¢ ces make it easier and more desirable for examiners to grant patents rather

than reject them (Farrell and Merges, 2004, IDEI report, 2006).

The patent quality problem raises many concerns particularly in the US.4 We may

ask, �rst: are bad quality patents harmful or not? Lemley (2001) claims that it is

reasonably e¢ cient to maintain a low standard of patent examination, in accordance

with the "rational ignorance principle". Speci�cally, he argues that the cost of a thor-

ough examination for each application would be prohibitive while inducing only a small

bene�t. Firstly, the majority of patents turn out to have insigni�cant market value im-

plying that the social cost of granting them is small even if they are invalid. Secondly,

if a weak but pro�table patent is granted, some market players will probably bring the

case before a court to settle the validity issue, if the patent is licensed at too high a

price.

These arguments have attracted much criticism. First, there are many reasons to

think that individual incentives to challenge a weak patent are rather low. A patentee

generally cares more about winning than a potential infringer does, since by winning

against a single challenger, a patentee establishes the validity of the patent against many

other potential infringers. By contrast, when infringers are competitors, a successful

challenge obtained by one of them bene�ts all (Farrell and Merges, 2004, Lemley and

Shapiro, 2005).5 Consequently, according to the free-riding argument, the individual

incentives to challenge a patent validity are weak. Moreover, according to the so-called

pass-through argument, licensees are induced to accept a high per-unit royalty rate

when they can decide to pass-on the royalty to their customers.6 Finally, an unsuccessful

attacker may be in jeopardy or even evicted from the market once deprived from the

overload leads to further deterioration of the examination process.4Europe is also concerned by the patent quality problem even though the post-grant opposition at

the EPO alleviates it (see Graham et al. 2003). The European situation in terms of patent quality isanalyzed in Guellec and von Pottelsberghe de la Potterie (2007) and the IDEI report (2006).

5Following the Blonder-Tongue decision (1971), it became clear that "the attacker is not able toexclude others from appropriating the bene�t of its successful patent attack", Blonder-Tongue Labs.,Inc. v. Univ. of Illinois Found, 402, U.S. 313, 350 (1971).

6"When multiple infringers compete in a product market, royalties are often passed-through, atleast in part, to consumers downstream. The pass-through will be stronger the more competitive theproduct market, the more symmetric the royalties, the more elastic the industry supply curve, and theless elastic the industry demand curve" (Farrell and Merges, 2004).

3

new technology, or required to pay a higher price than the licensees who have accepted

the licensing contract.

All these arguments suggest that individual incentives to challenge a patent may

be rather low. The probabilistic nature of patent protection and the low individual

incentives to challenge a patent may thus strengthen the market power of the licensor.

The owner of a probabilistic right and a potential user will come to a licensing agreement

as a private settlement to avoid the uncertainty of a court resolution. An agreement

bene�ts the holder of a weak patent while litigation and possible invalidation by a court

would deprive the licensor from any licensing revenue. However the licensing contract

will be accepted by the licensee only if its expected pro�t is at least as large as when

the patent validity is challenged. Therefore licensing an uncertain patent under the

shadow of patent litigation raises an interesting trade-o¤. We show in this paper that

di¤erent factors explain the issue of this trade-o¤: i) the nature of the licensing scheme

(per-unit royalty vs. up-front fee); ii) the patent�s strength measured by the probability

that it will be upheld; iii) the importance of the innovation; iv) the type of commitment

when dealing with an unsuccessful challenger; v) the possibility to engage in collective

negotiations of the licensing contract; vi) some market structure variables such as the

size of the downstream industry and the intensity of market competition.7

The literature on licensing and the properties of the di¤erent licensing mechanisms

has extensively examined the case of perfect patent protection. Based largely on pre-

vious works by Arrow (1962), Katz and Shapiro (1985, 1986), Kamien and Tauman

(1984, 1986), Kamien et al. (1992), the survey by Kamien (1992) summarizes the ma-

jor results, especially by comparing the patentholder�s pro�ts under di¤erent licensing

schemes. The patentee�s pro�ts are highest when licensing is made through an auction,

in which the patentee announces the number of licenses on o¤er and the latter accrue

to the highest bidders. The per-unit royalty scheme and the up-front fee mechanism

have been set against each other. While the earlier literature claimed that a per unit

royalty always generates lower pro�ts than a �xed fee, regardless of the industry size

and the magnitude of the innovation (Kamien and Tauman,1984 and 1986), a more

recent work has shown that when the number of �rms in the industry is su¢ ciently

high, the innovator�s payo¤ is higher with royalty licensing than with a �xed fee or

an auction (Sen, 2005). Moreover, some licensing methods induce full di¤usion, while

7Since a non-licensee su¤ers a negative externality when a competitor becomes a licensee, moreintense competition in the product market increases the licensor�s market power.

4

others lead to only partial di¤usion of the innovation: the number of licensees depends

on the licensing method and the magnitude of the cost reduction. In a more recent

contribution, Sen and Tauman (2007) generalize these �ndings by allowing the optimal

combination of an auction and a per-unit royalty in situations where the innovator may

be either an outsider or an insider in the downstream industry.8

Let us now consider how licensing is a¤ected when a patent is a probabilistic right.

Rough intuition suggests that licensing an uncertain patent in the shadow of patent

litigation leads to a license price which is proportional to the patent strength. This

intuition is not always correct for the following reason: when imperfect competition

occurs in the downstream industry, the free riding argument mentioned above lowers

the individual incentives to challenge the patent�s validity and this bene�ts the paten-

tholder. Farrell and Shapiro (2007) establishes two important properties for a minor

cost reducing innovation: (i) For weak patents, the royalty rate is as high as if the

patent were certain: it is equal to the magnitude of the cost reduction allowed by the

innovation; (ii) Whatever the patent strength, the royalty rate obtained in the shadow

of patent litigation exceeds the expected value of the royalty resulting from the patent

challenge. These strong properties have been obtained by considering a two-part li-

censing contract mechanism combining a per-unit royalty and a �xed fee, allowing for

instance a high royalty rate to be compensated by a negative transfer (i.e. an up-front

fee paid by the licensor to the licensee).9 Two restrictive assumptions have been used

in Farrell and Shapiro to obtain these results: �rst, they restrict their analysis to small

process innovations, i.e. innovations leading to a small cost reduction; second, they

assume that the best patentholder�s licensing strategy is to sell a license to all �rms

in the downstream industry, rather than to restrict the license supply to some �rms,

8Another burgeoning literature explores the consequences of informational asymmetries on licensing.Aoki and Hu (1996) examines how the choice between strategic licensing and litigating is a¤ected bythe levels of the litigation costs and their allocation between the plainti¤ and the defendant. Brocas(2006) identi�es two informational asymmetries: the moral hazard due to the inobservability of theinnovator�s R&D e¤ort, and the adverse selection due to the private value of holding a license. Macho-Stadler et al. (1996) introduces know-how transfer and shows that the patentholder prefers contractsbased on per-unit royalties rather than �xed fee payments. Other contributions, emphasizing eitherrisk aversion (Bousquet et al., 1998), strategic delegation (Saracho, 2002), strategic complementarity(Muto, 1993, Poddar and Sinha, 2004), or the size of the oligopoly market (Sen, 2005) reach the sameconclusion stating the superiority of the royalty licensing scheme.

9Farrell and Shapiro also investigate a two-part tari¤ in which the �xed fee is constrained to be nonnegative. However, in this case, their main result holds only under the two additional restrictions thatthe magnitude of the cost reduction innovation is small and all downstream �rms accept the licensingcontract at equilibrium.

5

leaving it to others to refuse and possibly initiate a litigation process.

In this paper we assess the robustness of these results by avoiding these assumptions,

and separately investigating two of the most common licensing mechanisms, namely the

per-unit royalty rate and the up-front fee. We analyze the properties of these mecha-

nisms, letting the licensor choose the number of licensees whatever the innovation size.

For both types of licensing schemes, we develop a three-stage game in which the paten-

tholder, acting as a Stackelberg leader, determines either a royalty rate or a �xed fee at

the �rst stage. At the second stage, each �rm independently decides whether to accept

the licensing contract. If it does not, it challenges the patent validity. If the patent

is found valid, the unsuccessful challenger is bound to use the old technology. If the

patent is found invalid, all the �rms in the oligopolistic industry have free access to

the technology. In the last stage, licensees and non-licensees compete in the product

market. Di¤erent variants of this basic model are examined in this paper, by intro-

ducing the possibility of a collective challenge or by allowing renegotiation between the

patentholder and an unsuccessful challenger.

Our paper departs from Farrell and Shapiro (2007) in several ways. First, unlike

Farrell and Shapiro who focus on a single licensing scheme combining a per-unit royalty

and a �xed fee, we separately investigate these two schemes; second, while they only

consider the case where the cost reduction is small, we investigate the consequences

of any cost reduction; third, we relax the crucial assumption of their paper stating

that the patentholder licenses every �rm in the industry, by endogeneizing the number

of licensees. We show below that this endogeneization has important consequences,

particularly when comparing the properties of the per-unit royalty rate and the up-

front fee licensing schemes. We also challenge the assumption that an unsuccessful

challenger is o¤ered a license at a price that captures its entire surplus.

We contribute to the literature on licensing uncertain patents on �ve points. First,

we show that while it is generally possible for the patentholder to reap some "extra

pro�t" by selling an uncertain patent under the per-unit royalty regime, the opportunity

to do so under the up-front fee regime disappears. This is due to the fact that the

patentee�s pro�t under a �xed fee regime is always equal to the expected pro�t in case

of litigation. Second, we show in the case of a linear demand under Cournot competition

that the patentee�s pro�ts may be higher with a per-unit royalty than with a �xed fee.

This result - which con�rms Sen (2005) - rests on a completely di¤erent argument based

on patent uncertainty. Third, for the per-unit royalty regime, we obtain necessary and

6

su¢ cient conditions under which the royalty rate resulting from a collective challenge is

lower than the expected royalty from an individual challenge. Fourth, we show that there

exist situations in which the per-unit royalty for a weak patent is below the expected

royalty in case of litigation. The latter result is obtained under general assumptions

on the pro�t functions and is con�rmed when post-trial renegotiation is introduced.

Finally, we show that the results obtained with perfect patents also hold when patents

are uncertain but strong: in this case, litigation never occurs.

The paper is organized as follows. Section 2 examines the per-unit royalty scheme.

It starts with the derivation of the maximum value of the per-unit royalty that deters

any litigation. This value is compared to two benchmarks: i/ the expected value of the

royalty in case of litigation; ii/ the royalty that would prevail under collective challenge

of the patent validity. The patentholder�s optimal royalty rate and its licensing revenues

are then determined. The conditions under which litigation is avoided at the subgame

perfect equilibrium are established. Section 3 analyzes the �xed fee licensing scheme. It

derives the demand for licenses and the licensing revenues as a function of the up-front

fee. These revenues are then compared to the expected revenues in case of litigation. In

Section 4, the two licensing mechanisms are compared from the licensor�s perspective.

Section 5 concludes by summarizing the results, putting them in an economic policy

perspective, and suggesting new research directions.

2 Royalty licensing schemes

In this section, we examine licensing schemes involving a pure royalty rate. More

precisely, we seek to determine the subgame perfect Nash equilibria of the following

three-stage game:

At the �rst stage the patentholder proposes a licensing contract by which a licensee

can use the new technology to reduce its marginal production cost from c to c�� againstthe payment of a per-unit royalty rate r:

At the second stage n �rms in a downstream industry simultaneously and indepen-

dently decide whether or not to purchase a license at the price r. If a �rm does not

accept the license o¤er, it can challenge the patent�s validity before a court. The out-

come of such a trial is uncertain: with probability � the patent is upheld by the court

and with probability 1 � � it is invalidated. The parameter � measures the patent�sstrength. If the patent is upheld, then a �rm that does not purchase the license uses

7

the old technology10 hence producing at marginal cost c whereas those who accepted

the license o¤er use the new technology and pay the royalty rate r to the patentholder,

having thus an e¤ective marginal cost equal to c� �+ r. If the patent is invalidated, allthe �rms, including those who accepted the o¤er can use for free the new technology

and their common marginal cost is c� �.At the third stage the downstream �rms compete in an oligopolistic product market.

The kind of competition that occurs is not speci�ed. It is simply assumed that there

exists a unique Nash equilibrium in the competition game between the members of the

oligopoly for any cost structure of the downstream �rms.

We sum-up the outcome of the third stage by denoting �(x; y) the equilibrium pro�t

function of an active �rm producing with marginal cost x while its (n� 1) competitorsproduce with marginal cost y: The case where �(x; y) = 0 is not excluded.

We assume the following general properties that are satis�ed by a large class of pro�t

functions (See Boone, 2001, and Amir and Wooders, 2000).

A1. The equilibrium pro�t function �(x; y) is continuous in both it arguments over

[c� �; c] � [c� �; c] and twice di¤erentiable in both its arguments over the subset of[c� �; c]� [c� �; c] in which �(x; y) > 0:A2. The equilibrium pro�t of a �rm is decreasing in its own cost : if �(x; y) > 0

then �1(x; y) < 0 and if �(x; y) = 0 then �(x0; y) = 0 for any x0 � x:A3. The equilibrium pro�t of a �rm is increasing in its competitors� costs : if

�(x; y) > 0 then �2(x; y) > 0 and if �(x; y) = 0 then �(x; y0) = 0 for any y0 � y:A4. In a symmetric oligopoly, an identical drop in all �rms�costs raises each �rm�s

pro�t: �1(x; x) + �2(x; x) < 0:

Given A2 and A3, A4 means that own cost e¤ects dominate rival�s cost e¤ects.

The subgame perfect Nash equilibria of the game are obtained as usual by backward

induction.10This assumption may seem quite strong but recall that IP laws do not compel patentholders to

license others, particularly those who challenge the validity of a patent or sue the patentholder forinfringement of their own patents. To illustrate, when Intergraph (a company producing graphic workstations) sued Intel (micro-processors) for infringement of its Central Processing Unit patent, Intelcountered by removing Intergraph from its list of customers and threatening to discontinue the sale ofIntel microprocessors to Intergraph (See Encaoua and Hollander, 2002). We relax later this assumptionby introducing renegotiation between the unsuccessful challenger and the patentholder.

8

2.1 Accepting or not the patentholder�s o¤er: second stage

We start by determining the set of royalty rates r such that a Nash equilibrium leads

to an outcome in which every downstream �rm accepts such a royalty. This occurs if

and only if no single �rm has an incentive to deviate by refusing to buy a license at this

rate and challenging the patent�s validity. Since an unsuccessful challenger produces at

cost c while its competitors that have accepted the licensing contract produce at cost

c� �+ r; a per-unit royalty rate r that is accepted by every �rm at equilibrium satis�esthe condition:

�(c� �+ r; c� �+ r) � ��(c; c� �+ r) + (1� �)�(c� �; c� �) (1)

The following lemma characterizes the set of royalty rates that satisfy inequality (1).

Lemma 1 A royalty rate r is accepted by all �rms if and only if r � r (�) where

r (�) 2 [0; �] is the unique solution in r to the equation �(c � � + r; c � � + r) =��(c; c� �+ r) + (1� �)�(c� �; c� �)

Proof. See Appendix.When analyzing the maximum value of the per-unit royalty such that all �rms accept

the contract, two cases emerge.

Case 1: the magnitude � of the cost reduction is such that �(c; c� �) = 0.This case occurs for a su¢ ciently large innovation (high value of �) or for a su¢ ciently in-

tense competition (e.g. large number n of downstream �rms). In such a case, according

to assumptions A1 and A3, there exists a threshold r 2 [0; �] such that �(c; c��+r) = 0if r � r and �(c; c� �+ r) > 0 if r > r. An unsuccessful challenger will get zero pro�tif the royalty rate is below the threshold (r � r), and a positive pro�t if the royalty

rate is above the threshold (r > r) :

First consider a contract involving a royalty rate r � r: According to condition (1), itwill induce a Nash equilibrium where all the �rms will accept the licensing contract if

and only if:

�(c� �+ r; c� �+ r) � (1� �)�(c� �; c� �) (2)

Denote r2(�) the solution in r to the equation �(c��+r; c��+r) = (1��)�(c��; c��):It is easy to show that inequality (2) is equivalent to r � r2 (�) :

9

Second consider a contract involving a royalty rate r > br: It will be accepted by all �rmsif and only if inequality (1) is satis�ed with �(c; c��+r) > 0: Denote r1 (�) the solutionto the equation in r �(c� �+ r; c� �+ r) = ��(c; c� �+ r)+ (1� �)�(c� �; c� �) whenthis solution is greater than r. Denote by � 2 [0; 1] the unique solution to the equationr2 (�) = r: The existence of � 2 [0; 1] can be derived from the two following properties:

i/ r2 (0) = 0 � r and r2 (1) = � � r; ii/ r2 (:) is continuous over [0; �]. Its uniquenessfollows from the strict monotonicity of r2 (:) : Note that r1

���= r2

���= r:

Summing-up these possibilities, a royalty rate r will be accepted by all �rms if the

following condition holds:

r � min(r; r2(�)) or r � r � r1 (�)

Note that if � � �, the previous condition is equivalent to r � r2(�); and if � > �; it isequivalent to r � r1 (�) : This means that the maximum royalty rate inducing a Nash

equilibrium where all �rms accept the license o¤er is given by:

r (�) =

(r2(�) if � � �r1 (�) if � > �

Case 2: the magnitude � of the cost reduction is such that �(c; c� �) > 0In this case, whatever the royalty rate �xed by the patentholder, the pro�t of a �rm

challenging the patent�s validity remains positive: �(c; c � � + r) � �(c; c � �) > 0:

This implies that the equilibrium value r (�) of the per-unit royalty that makes all �rms

accept the contract is equal to r1 (�) : In this case r (�) = r1 (�) for all � 2 [0; 1] :

2.1.1 Royalty rate benchmarks

Having characterized the per-unit royalty level r(�), it is interesting to compare it to

two benchmarks: i/ the expected value of the maximum royalty rate in case of litigation

denoted re(�); ii/ the royalty rate deterring a collective challenge denoted rc(�).

First benchmark: the expected value of the maximum royalty rate in case of litigation.

This benchmark is easily computed: with probability � the patent is upheld by the court,

hence becoming an ironclad right that can be licensed at a maximum per-unit royalty

r(1) = �; and with probability 1�� the patent is invalidated and the �rms can use it for

10

free, leaving the patentholder with a royalty r(0) = 0. Therefore, the expected value of

the maximum royalty rate in case of litigation is equal to re(�) = �r(1)+(1��)r(0) = ��.The expected value of the maximum royalty in case of litigation is thus proportional to

the patent�s strength �. This benchmark is interpreted in Farrell and Shapiro (2007) as

the ex ante value of the per-unit royalty rate that an applicant of a process innovation

reducing the cost by � can expect when the patent has a probability � to be granted by

the patent o¢ ce.

Second benchmark: the royalty rate deterring a collective challenge.

Suppose that at stage 2 the �rms cooperatively agree on whether buying the license or

challenging all together the patent�s validity. In this case, the �rms will cooperatively

accept a licensing contract involving a royalty rate r if and only if:

� (c� �+ r; c� �+ r) � �� (c; c) + (1� �)� (c� �; c� �)

The functionw de�ned byw(r) = � (c� �+ r; c� �+ r)��� (c; c)�(1� �)� (c� �; c� �)is continuous, strictly decreasing (A3) and satis�es the conditions w (0) � 0 and

w (�) � 0: Hence there exists a unique solution rc(�) 2 [0; �] to the equation w (r) = 0,and the inequality w (r) � 0 is equivalent to r � rc(�): All �rms cooperatively acceptto buy a license at a royalty rate r if and only if r � rc(�). Some properties of this

second benchmark rc(�) are easily obtained.

Proposition 2 The function rc(�) is increasing. (i) It is concave over [0; 1] if and onlyif the function x ! �(x; x) is concave over [c� �; c]. In this case rc(�) � re(�) = ��

for all � 2 [0; 1]; (ii) It is convex over [0; 1] if and only if the function x ! �(x; x) is

convex over [c� �; c]. In this case rc(�) � re(�) = �� for all � 2 [0; 1] :

Proof. See Appendix.The convexity of the equilibrium pro�t function �(x; x) is satis�ed for di¤erent demand

speci�cations including for instance a linear demand and a Cournot behavior, while

it is di¢ cult, if not impossible, to �nd a speci�cation of the demand function leading

to a concave equilibrium pro�t function �(x; x).11 This suggests that the inequality

rc(�) � �� is more likely satis�ed than the reverse one. Thus the royalty rate deterring11With a linear demand funtion Q = a � p, a marginal cost x; and an oligopoly of n �rms, the

Cournot pro�t equilibrium is �(x; x) = (a�x)2(n+1)2 which is a convex function of x.

11

a collective challenge (rc(�)) is likely to be lower than the expected royalty rate in case

of an individual challenge (re(�)).

2.1.2 Comparison of r(�) to re(�) = ��

Analyzing the shape of the function � ! r(�) allows to compare the per-unit royalty

rate r(�) that deters individual challenge to the benchmark re(�) = �� which represents

the expected royalty rate in case of individual litigation:

Recall �rst that when the innovation � is such that � (c; c� �) = 0; we have r (�) =r2(�) over the interval

h0; �i: It is easy to show that r2(�) is increasing in �: Indeed,

di¤erentiating with respect to � the equation �(c��+r2(�); c��+r2(�)) = (1��)�(c��; c� �) we get:

r02(�) =��(c� �; c� �)

(�1 + �2)(c� �+ r2(�); c� �+ r2(�))which implies that r02(�) > 0 since �1+ �2 < 0 (A4): Therefore r2(�) increases with the

patent strength �:

Furthermore, we can derive some properties about the monotonicity of r02(�) and conse-

quently about the convexity or concavity of r2(�): Note that (�1+�2)(c� �+ r2(�); c��+ r2(�)) is increasing (resp. decreasing) in � if (�1 + �2) (x; x) is increasing (resp. de-

creasing) in x; which is equivalent to � (x; x) convex (resp. concave) in x. Hence r2(�)

is convex (resp. concave) over the intervalh0; �iif � (x; x) is convex (resp. concave) in

x.

We can also compare r02(0) to �. This comparison matters when comparing r(�) = r2(�)

to the benchmark re(�) = �� for small values of � (weak patent). Indeed, if r02(0) > �

(resp. r02(0) < �) then for � su¢ ciently small we will have r2(�) > �� (resp. r2(�) < ��).

Since r2(0) = 0, we have:

r02(0) =��(c� �; c� �)

(�1 + �2)(c� �; c� �)

Therefore,

r02(0) > �()��(c� �; c� �)

�(�1 + �2)(c� �; c� �)> 1

Denoting �(�) = �(c� �; c� �), we obtain:

r02(0) > �()�(�)

��0 (�)> 1() � (�) < 1

12

where � (�) = ��0(�)�(�)

= �n�0(�)n�(�)

is the elasticity of the industry pro�ts with respect to a

cost reduction � in the marginal cost of all the industry�s �rms. These results lead to

the following proposition:

Proposition 3 If � (x; x) is concave in x over [c� �; c] then r2(�) is concave overh0; �i

and r(�) = r2(�) � �� for any � 2h0; �i

If � (x; x) is convex in x over [c� �; c] then r(�) = r2(�) is convex overh0; �iand

the location of r2(�) with respect to �� depends on � (�):

- if � (�) < 1 then r2(�) � �� for any � 2h0; �i

- if � (�) > 1 then there exists �� such that r2(�) < �� for 0 < � < �� and r2(�) � ��for � � ��:

Since the equilibrium pro�t function � (x; x) is more likely to be convex than concave in

x, the per-unit royalty rate r2(�) is more likely to have a convex shape for small values

of �: In this case, the maximum royalty rate r2(�) that deters individual challenge may

be lower than the expected royalty re(�) = �� for weak patents (small value of �) if

the industry pro�ts are elastic with respect to � (i.e. � (�) > 1). We illustrate this

possibility in the following example.

Example: Consider a Cournot oligopoly with a constant-elasticity demand func-tion: D(p) = p�

1� where � < 1. It is straightforward to show that the equilibrium

pro�t of a �rm in a symmetric oligopoly with marginal cost c is given by: �(c; c) =�n��

�n��n1��

� 1� c1�

1� which is convex in c when � < 1: The elasticity of the industry pro�ts

with respect to � is given by � (�) = �c��

�1�� 1�. Therefore, � (�) < 1 () � < �c:

Hence for "major innovations", i.e. innovations such that �c < � < c, the royalty rate

r2(�) is lower than the benchmark level re(�) = �� for "weak patents"�� < ��

�:

Let us now turn to the properties of r1(�). Recall that r(�) = r1(�) overi�; 1iif

�(c; c� �) = 0 and r(�) = r1(�) over [0; 1] if �(c; c� �) > 0. By de�nition of r1(�), wehave:

�(c� �+ r1(�); c� �+ r1(�)) = ��(c; c� �+ r1(�)) + (1� �)�(c� �; c� �)

Di¤erentiating this equation with respect to � we get:

r01(�) =�(c; c� �+ r1(�))� �(c� �; c� �)

(�1 + �2)(c� �+ r1(�); c� �+ r1(�))� ��2(c; c� �+ r1(�))(3)

13

We have �(c; c � � + r1(�)) � �(c � � + r1(�); c � � + r1(�)) < �(c � �; c � �): The�rst inequality follows from r1(�) � � and the second one from �1(x; x) + �2(x; x) < 0:

Hence the numerator in (3) is negative. The denominator is negative as well since

�1(x; x) + �2(x; x) < 0 and �2(x; y) > 0: Consequently r01(�) > 0; that is r1(�) is

increasing in the patent strength �:

We can derive the position of r1(�) relative to re (�) = �� for � su¢ ciently close to 1,

i.e. su¢ ciently strong patents, from the comparison of r01(1) to �: Note that r01(1) =

�(c;c)��(c��;c��)�1(c;c)

: Therefore if the slope �(c;c)��(c��;c��)�

is strictly greater (resp. smaller)

than the negative partial derivative �1(c; c) than r1(�) < �� (resp. r1(�) > ��) for

su¢ ciently strong patents.

2.1.3 Comparison of r(�) to rc(�)

The e¤ect of free-riding is measured by the di¤erence r(�)� rc(�): It is easy to see thatthis di¤erence is positive. This follows from the fact that � (c; c) � � (c; c� �+ r) forany r � 0: This will in particular be true for r = rc(�): Since � (c� �+ rc(�); c� �+ rc(�)) =�� (c; c)+(1� �)� (c� �; c� �) we obtain that � (c� �+ rc(�); c� �+ rc(�)) � �� (c; c� �+ rc(�))+(1� �)� (c� �; c� �) : This last inequality implies that a royalty rate r = rc(�) will benon cooperatively accepted by all �rms if proposed by the patentholder. Therefore

rc(�) � r(�) for all � 2 [0; 1] : The public good nature of the challenge implies that themaximum royalty rate that the patentholder can obtain in the licensing of an uncer-

tain patent is higher under individual challenge (r(�)) than under collective challenge

(rc(�)).

Moreover, if �(x; x) is convex in x over [c� �; c], then rc(�) � �� for all � 2 [0; 1]

whereas r (�) may be above or below than ��: For instance if �(�) < 1 then r(�) � ��for � 2

h0; �i(see Lemma 3) while if �(�) > 1 then r(�) � �� for � su¢ ciently small�

� < min��; ����:We show in subsection 2.4 that with Cournot competition and linear

demand, r (�) is above �� while rc(�) is below ��: rc(�) < re(�) = �� < r(�):

All these results are summarized in Figure 1 which represents four possible shapes

of r(�) relative to the expected royalty re(�) in case of litigation (represented by the

straight line � ! ��).

14

Figure 1: Possible shapes of r (�) compared to re (�) = ��

2.1.4 The second stage equilibria

The following proposition gives a complete characterization of the possible second stage

equilibria.

Proposition 4 For a patentholder�s o¤er involving a royalty rate r, the equilibria ofthe second stage are as follows: i/ if r � r(�) then the unique equilibrium is given by all�rms accepting the license o¤er; ii/ if r(�) < r � � all the equilibria involve a number of(n� 1) license buyers; iii/ if r > � the unique equilibrium is given by all �rms refusing

the license o¤er.

Proof. See AppendixThis proposition states that two possibilities are o¤ered to a holder of an uncertain

patent with strength � when selling licenses through a per-unit royalty rate: either

the royalty r is chosen just equal to the maximal value r(�) that deters any challenge,

15

and in this case n licenses are sold, or the chosen royalty rate r is above this value

(r(�) < r � �), and in this case one and only one �rm challenges the patent validity

(n� 1 licenses are sold).

2.2 The patentholder�s optimal license o¤er: �rst stage

We turn now to the patentholder�s optimal decision at the �rst stage of the game.

Denote q(c � � + r; k) the individual output of a licensee when the per-unit rate r isaccepted by k �rms, and the n� k remaining �rms produce at marginal cost c.The patentholder�s licensing expected revenues P (r) are given by

P (r) =

8><>:nrq(c� �+ r; n) if r � r(�)

�(n� 1)rq(c� �+ r; n� 1) if r(�) < r � �0 if r > �

Note that when r 2]r(�); �]; one �rm refuses to buy a license and challenges the patent

validity and the other (n� 1) �rms buy a license (proposition 5). Therefore the paten-tholder�s licensing expected revenues depend on the issue of the trial (the patent is

upheld with probability �).

Let us introduce the following assumptions:

A5. A licensee�s output is nonincreasing in the number of licenses: q(c��+r; n�1) �q(c� �+ r; n) for all r 2 [0; �] :A6. The aggregate output is nondecreasing in the number of licenses: Q(c � � +

r; n) � Q(c� �+ r; n� 1) for all r 2 [0; �] :A7. The function krq (c� "+ r; k) is concave in r for k 2 fn� 1; ng :

Denote ~rk (�) = argmaxr�0

krq (c� "+ r; k) for k 2 fn� 1; ng : As a function of the royalty

rate the licensing revenue is a concave function (A7) that reaches its maximum at the

value ~rk (�) when k licenses are sold.

In order to determine the maximum of P (r) over [0; r(�)] and [r(�); �] ; we need to

compare � and ~rk (�). To do so, we must distinguish between di¤erent cases according

to the location of " with respect to ~rn�1 (�) and ~rn (�).

The following lemma is useful for the subsequent analysis:

Lemma 5 If � � ~rn�1 (�) then � � ~rn (�) :

Proof. See Appendix

16

A straightforward consequence of the lemma is that if � > ~rn (�) then � > ~rn�1 (�) as

well. Therefore, only three cases must be investigated: i/ � � ~rn�1 (�); ii/ ~rn�1 (�) <

� � ~rn (�); iii/ � > ~rn (�) :The following propositions determine the patentholder�s optimal choice r�(�) in each of

these cases and identify the conditions under which litigation is deterred at the subgame

perfect equilibrium.

Proposition 6 If � � ~rn�1 (�) ; the function s(�) de�ned as the unique solution in r tothe equation nrq(c� �+ r; n) = �(n� 1)�q(c; n) is convex over [0; 1], satis�es s(0) = 0,s(1) < �; and the per-unit royalty that maximizes the licensing revenues is given by

r� (�) =

(r (�) if r (�) � s (�)� if r (�) < s (�)

In this case, litigation is deterred at equilibrium if and only if r (�) � s (�)

Proof. See AppendixThis proposition characterizes the optimal royalty rate for the patentholder when the

magnitude � of the cost reduction is such that � � ~rn�1 (�). First, the function s(�)

de�nes the royalty rate level for which the patentholder is indi¤erent between selling

n licenses at the price r(�) and selling (n � 1) licenses at the higher price � (in whichcase litigation occurs and the expected licensing revenues are �(n � 1)�q(c; n)). Notethat when � � ~rn�1 (�) ; if the license is sold to only (n� 1) �rms, the optimal royaltyrate is � because the licensing revenue is an increasing concave function of r over [0; �].

Second, the comparison between the maximum rate r(�) satisfying equation (1) and

the royalty rate s(�) leads to the following decision: if r (�) � s (�) it is optimal to

set r�(�) = r(�) and this choice deters litigation; if r (�) < s (�) it is optimal to set

a higher price r�(�) = � and let one �rm challenge the patent validity. Note that if

r(�) is convex and the curves r(�) and s (�) meet in only one point over ]0; 1[ ; then

the curve r(�) necessarily intersects the curve s(�) from below since r(0) = s(0) = 0

and s(1) < r(1) = �: This implies that for low values of �, we have r (�) < s (�) and

the optimal per-unit royalty rate is then independent of � and is the same as if the

patent were certain. The same result appears in Farrell and Shapiro (2007) but the

justi�cation is di¤erent here. While Farrell and Shapiro consider only the case where

the cost reduction magnitude � is small enough and assume that all �rms buy a license

17

at equilibrium, we obtain the same result by allowing the number of licensees to depend

on the per-unit royalty. It is precisely when the royalty at which all �rms accept to buy

a license is too low (i.e. r(�) < s(�)) that the holder of a weak patent prefers to sell it

at the higher price �, triggering thus a patent litigation.

We turn now to the second case where ~rn�1 (�) < � � ~rn (�) :

Proposition 7 If ~rn�1 (�) < � � ~rn (�) ; then de�ning v (�) as the unique solution in rto the equation nrq(c� � + r; n) = (n� 1)�~rn�1 (�) q(c� � + ~rn�1 (�) ; n� 1); and ~�n�1as the solution to the equation r(�) = ~rn�1 (�) ; the function v(�) is convex over [0; 1],

v(0) = 0, v(1) < �; and we have

r� (�) =

(~rn�1 (�) if � < ~�n�1 and r (�) < v (�)

r (�) otherwise

In this case, litigation is deterred at equilibrium if and only if at least one of the two

following conditions hold: � � ~�n�1 or r (�) � v (�)

Proof. See Appendix.To interpret this proposition, one must �rst note that if the patentholder �nds it optimal

to trigger a litigation by selling at a royalty r > r(�), the optimal royalty rate is given

by ~rn�1 (�) since ~rn�1 (�) < �: The expected licensing revenues are therefore equal to

(n�1)�~rn�1 (�) q(c� �+~rn�1 (�) ; n�1): The function v(�) de�nes the royalty rate levelfor which the patentholder is indi¤erent between selling n licenses at the price r(�) and

selling (n � 1) licenses at the price ~rn�1 (�) : Second, it is optimal to sell only (n � 1)licenses at the per-unit royalty ~rn�1 (�) as long as v(�) > r(�) and � < ~�n�1 where ~�n�1is the solution to the equation r(�) = ~rn�1 (�) : This means that the holder of a weak

patent (� < ~�n�1) prefers to trigger a patent litigation by selling licenses at a per-unit

royalty rate ~rn�1 (�) when the royalty that all the �rms accept is too low (r(�) < v(�)).

Again, this extends the result obtained by Farrell and Shapiro (2007) in the sense that

the optimal per-unit royalty rate r� (�) for a weak patent (� < ~�n�1) is independent

of the patent strength � and is the same as if the patent were certain (i.e. r� (�) =

~rn�1 (�)). In our model, it is because the per-unit royalty accepted by all the �rms for

a weak patent is too low that the patentholder prefers to sell at the royalty rate that

maximizes its pro�t as if the patent were certain, triggering thus a patent litigation.

The last case occurs when � > ~rn (�)

18

Proposition 8 If � > ~rn (�) then, de�ning ~�n as the unique solution to the equation

r(�) = ~rn (�) ; we have

r� (�) =

8>><>>:~rn�1 (�) if � � min

�~�n�1; ~�n

�and r (�) < v (�)

~rn (�) if � � ~�n�1r (�) otherwise

In this case, litigation is deterred at equilibrium if and only if at least one of the two

following conditions hold: � > min(~�n�1; ~�n) or r (�) � v (�) :

Proof. See Appendix.The interpretation is the same as in the two previous propositions except that for

� > ~rn (�) (implying that � > ern�1(�)), when it is optimal for the patentholder to triggera litigation by selling at a royalty r > r(�), the optimal royalty rate is given either by

~rn�1 (�) or ~rn (�). Again the optimal per-unit rate of a weak patent is the same as if

the patent were certain, but in this case a patent litigation does not necessarily occur

when r� (�) = ~rn (�) :

The following corollary gives a su¢ cient condition for litigation deterrence.

Corollary 9 If r (�) > �� then the patentholder �nds it optimal to deter litigation andP (r� (�)) > �P (r� (1)):

Proof. See Appendix.This corollary states that if the maximum royalty rate r(�) acceptable by all �rms

is above the expected value of the royalty in case of litigation re(�) = ��, then the

patentholder will prefer to deter litigation. It gives thus a su¢ cient condition under

which the patentholder takes advantage of both the uncertainty of its patent and the

externalities between the downstream competitors. The consequence is clear. Insofar as

the maximum royalty rate that deters litigation is higher than the expected royalty in

case of litigation, the patentholder gets a higher pro�t than the ex ante expected pro�t

it could get if the patent were granted by the patent o¢ ce with the same probability

that the court upholds the patent validity: the per-unit royalty licensing scheme gives

to the patentholder a pro�t P (r� (�)) that is higher than the expected pro�t �P (r� (1))

resulting from the uncertainty on the patent validity. The corollary fully justi�es the

use of the benchmark re(�) = ��:

19

2.3 Introduction of renegotiation

So far we have assumed that in case of litigation, an unsuccessful challenger produces

with marginal cost c because the patentholder refuses to sell him a license. Whether

such a commitment to refuse a license to an unsucessful challenger is credible or not

must be discussed. From the challenger�s perspective this commitment is equivalent to

an o¤er of a new licensing contract involving a royalty rate �r = �. However, from the

patentholder�s perspective, this equivalence does not hold. Moreover a situation where

an unsuccessful challenger is o¤ered a new licensing contract involving a royalty rate

�r < � may be preferred to a license refusal: Such an issue is important since a potential

challenger will take the decision whether to accept the license or contest the patent�s

validity, anticipating what will happen if the patent is validated.

Formally if we allow for renegotiation when (n � 1) �rms accept a licensing contractbased on a royalty rate r and the remaining �rm challenges the patent unsuccessfully,

then the patentholder will o¤er to the challenger a contract involving a royalty rate

�r 2 [0; �] that maximizes its licensing revenues

P (r; �r) = (n� 1) rqL (c� �+ r; c� �+ �r) + �rqNL (c� �+ r; c� �+ �r)

where qL (c� �+ r; c� �+ �r) denotes the equilibrium quantity produced by each of

the (n� 1) �rms that accepted initially the license o¤er r and qNL (c� �+ r; c� �+ �r)is the equilibrium quantity produced by the unsuccessful challenger who produces at

marginal cost c� �+�r: If �r(r) is the royalty rate that maximizes P (r; �r) with respect to�r, a licensing contract involving a royalty rate r will be accepted by all the downstream

�rms if and only if:

�(c� �+ r; c� �+ r) � ��(c� �+ �r (r) ; c� �+ r) + (1� �)�(c� �; c� �) (4)

Since �r (r) � � we have �(c � � + �r (r) ; c � � + r) � �(c; c � � + r) which entails thatconstraint (4) is (weakly) more stringent than (1). More precisely, a royalty rate r could

be accepted if the patentholder commits to refuse a license to a challenger or license him

at �r = �, but not accepted if he cannot commit. This implies that the maximum royalty

rate the patentholder can make the n �rms pay is (weakly) smaller when renegotiation

of a licensing contract (after patent validation) is introduced. This is illustrated in the

case of Cournot competition with linear demand.

20

2.4 Cournot competition with linear demand

Assume that the downstream �rms compete à la Cournot in an homogeneous market

where the demand is given by Q = max(a � p; 0) where a > c: If x is the constant

marginal cost of a �rm and y � x the (common) marginal cost of the remaining (n� 1)�rms, denote by �(x; y) the equilibrium pro�t of the �rm with a cost x when confronted

to (n� 1) competitors with a cost y. The �rm having a higher marginal cost x > y is

active on the market if and only if x < a+(n�1)yn

:When this condition is met, we obtain

�(x; y) = (a�nx+(n�1)y)2(n+1)2

. From this expression we derive: �1(x; y) = � 2nn+1

a�nx+(n�1)yn+1

<

0 ; �2(x; y) =2(n�1)n+1

a�nx+(n�1)yn+1

> 0 ; �1(x; y) + �2(x; y) = � 2n+1

a�nx+(n�1)yn+1

< 0 ;

�11(x; y) =2n2

(n+1)2> 0 ; �12(x; y) = �2n(n�1)

(n+1)2< 0 ; �22(x; y) =

2(n�1)2(n+1)2

> 0: Note that

in this case the function x ! � (x; x) is convex (@2�(x;x)@x2

= �11(x; x) + 2�12(x; x) +

�22(x; x) =2

(n+1)2> 0):

2.4.1 Determination of the acceptable royalty rates

When � < a�cn�1 ; we have �(c; c� �) > 0: Therefore, if � <

a�cn�1 , a licensing contract with

a royalty rate r is accepted by all �rms if and only if r � r1(�) where r1(�) is the uniquepositive solution in r to equation (1) which, in the case of Cournot competition with

linear demand, is equivalent to the following equation:

(a� c+ �� r)2 = �[a� c� (n� 1)(�� r)]2 + (1� �)(a� c+ �)2 (5)

When � � a�cn�1 ; we have �(c; c � �) = 0. We determine the value br such that the

inequality �(c; c � � + r) > 0 is equivalent to r > br: A simple calculation leads tobr = � � a�cn�1 : Therefore, a licensing contract based on a royalty rate r � br is accepted

if and only if r � r2(�) where r2(�) is the unique solution in r 2 [0; �] to the followingequation:

[a� c+ �� r]2 = (1� �)[a� c+ �]2

The positive solution of this equation is given by r2(�) = [1�p1� �](a� c+ �): This

expression can be used to determine the patent strength threshold � = 1� [ n(a�c)(n�1)(a�c+�) ]

2

such that r2(�) = r:

Thus, if � � a�cn�1 ; the maximum royalty rate the patentholder can make all the down-

stream �rms accept is r(�) = r1(�) for all � 2 [0; 1], while if � � a�cn�1 , the maximum

21

royalty rate the patentholder can make all the downstream �rms accept is given by:

r(�) =

(r2(�) = [1�

p1� �](a� c+ �) if 0 � � � �r1(�) if � � � � 1

Note that the royalty rate r2(�) = [1 �p1� �](a � c + �) is convex in � and that

r02(0) =12(a � c + �) > � for any non-drastic innovation, i.e. � � a � c: Since r2(�) is

convex overh0; �i, we can state that r2(�) � r2(0) + �r02(0) = �r02(0) which entails that

r(�) = r2(�) � �� for � � a � c and � 2h0; �i. Thus, the maximum royalty rate r(�)

acceptable by all �rms is higher than the benchmark value re(�) = �� for � � �:12

2.4.2 Optimal choice of the royalty rate by the patentholder

Consider an innovation � � a�cn�1 covered by a patent of strength � � �: The paten-

tholder�s licensing revenues when the n �rms accept to pay a royalty rate r are equal to

nrq(c��+r; n) = nr (a�c+��r)n+1

, which is a concave function in r. Note that the condition

� � ~rn (�) where ~rn (�) = argmax[nrq(c��+r; n)] = a�c+�2

is equivalent to � � a�c; i.e.the innovation is non-drastic. Moreover, the maximum royalty rate when (n� 1) �rmsbuy the license is given by ~rn�1 (�) = argmax[�(n� 1)rq(c� �+ r; n� 1)] = a�c+n�

2n; so

that in the linear case we have ~rn (�) > ~rn�1 (�) for any � > 0: Note also that r(�) is con-

vex over � 2 [0; 1] because �(x; x) is convex in x over [c� �; c] : Since r (�) = r2(�) � ��for � 2

h0; �i; corollary (9) and its proof entail that r� (�) = min (r (�) ; ~rn (�)) and

litigation does not occur in this case.

The patentholder�s royalty rate choice can be described as follows.

- For an innovation � such that a�cn< � � a� c, we have � � ~rn (�) : Since r (�) � �

then r (�) � ~rn (�) and consequently r�(�) = r (�) = [1 �p1� �](a � c + �) for all

� 2h0; �i: the royalty rate the patentholder will set is equal to the maximum royalty

rate that all downstream �rms accept.

- For an innovation � such that � > a � c, we have � > ~rn (�) : Consequently, for

such a drastic innovation, the patentholder will choose to set the royalty rate to the

value r(�) if and only if this value does not exceed ~rn (�) : The unique solution in � to

the equation r(�) = ~rn (�) is given by e�n = 34. Note that e�n < � is true if and only if

� > n+1n�1 (a� c) : Thus two subcases emerge:

12The linear demand case illustrates thus proposition 3 in the case where �(x; x) is convex and�(�) < 1:

22

i/ if a � c < � � n+1n�1 (a� c) then the patentholder�s optimal choice of royalty rate

is given by r�(�) = r (�) = [1�p1� �](a� c+ �) for all � 2

h0; �i;

ii/ if � > n+1n�1 (a� c) then

r�(�) =

([1�

p1� �](a� c+ �) if 0 � � � 3

4a�c+�2

if 34� � � �

2.4.3 Renegotiation

Suppose now that the possibility to renegotiate a licensing contract with an unsuccessful

challenger is introduced. Denote �rm n the challenging �rm and �r the per-unit royalty

rate at which a license is o¤ered if the challenge fails. Cournot competition between

(n � 1) �rms (indexed by i = 1; 2; :::; n � 1) whose marginal cost is c � � + r and �rmn whose marginal cost is c� �+ �r leads to the following equilibrium outputs:

qi(r; �r) =

(a�c+��2r+�r

n+1if i = 1; ::; n� 1

a�c+��n�r+(n�1)rn+1

if i = n

For a given r, the value of the royalty rate �r that maximizes the patentholder�s licensing

revenue is the solution to the following program:

maxr2[0;�]

P (r; �r) = (n� 1)ra� c+ �� 2r + �rn+ 1

+ �ra� c+ �� n�r + (n� 1)r

n+ 1

Suppose that the innovation is non-drastic, i.e. � < a � c: The unique unconstrainedmaximum of the concave function �r �! P (r; �r) is given by the FOC @P (r;�r)

@�r= a�c+�+2(n�1)r�2n�r

n+1=

0. The maximum of the function P (r; �r) over the interval �r 2 [0; �] is reached at

�r(r) = min

��;(n� 1)n

r +a� c+ �2n

�= min

��; r +

1

n

�a� c+ �

2� r��

Since � < a � c, we have a�c+�2

> �. Therefore, r 2 [0; �] =) a�c+�2

� r � 0 and

consequently �r(r) � r:Hence a �rmwhich refuses a licensing contract and unsuccessfullychallenges the patent�s validity will get a new licensing o¤er with a higher royalty rate

than the royalty paid by licensees that have accepted the initial licensing contract.13

Moreover, the condition �r(r) < � is ful�lled if and only if r < ( 2n�12(n�1))� �

a�c2(n�1) � ';

13 It is obvious that the patentholder�s position is stronger after the patent has been upheld by thecourt than before.

23

which is positive whenever � > a�c2n�1 : For such a royalty rate r, we have �(c � � +

�r(r); c� �+ r) = [a�c+��n�r(r)+(n�1)rn+1

]2, and the condition expressing that all �rms accept

the licensing contract r is:

�(c� �+ r; c� �+ r) � ��(c� �+ �r(r); c� �+ r) + (1� �)�(c� �; c� �)

Replacing �r(r) by its value, one obtains:

(a� c+ �� r)2(n+ 1)2

� ��a� c+ �2(n+ 1)

�2+ (1� �)

�a� c+ �(n+ 1)

�2=4� 3�4

�a� c+ �(n+ 1)

�2This inequality is satis�ed if and only if:

r � (a� c+ �)�1�

p4� 3�2

�Hence a royalty rate r < ' is accepted by all �rms if and only if the previous inequality

holds. Denoting �� the unique solution in � to the equation (a� c+ �)�1�

p4�3�2

�= ';

we can then state that for � � ��; the maximum royalty rate accepted by all �rms whenpost-trial license o¤er is possible is given by:

rp (�) = (a� c+ �)�1�

p4� 3�2

�Straightforward computations lead to drp

d�(0) = 3

8(a � c + �): It is easy to show that

drp

d�(0) < � for any � 2

�35(a� c) ; a� c

�. Consequently for such intermediate innova-

tions, rp (�) < �� for su¢ ciently small values of �: Note that for such innovations, the

condition � > a�c2n�1 is satis�ed since

35(a� c) > a�c

2n�1 for any n � 2:These results are summarized in the following proposition:

Proposition 10 In a Cournot model with an homogeneous product and a linear de-mand Q = max(a � p; 0) the maximum per-unit royalty rate that induces a unique

perfect subgame equilibrium in which all �rms choose to buy a license of a patented

technology that reduces the marginal cost by � 2�35(a� c) ; a� c

�is given by rp (�) =

(a� c + �)�1�

p4�3�2

�for a patent strength � smaller than a threshold � 2]0; 1[. The

royalty rp(�) is sustained by a renegotiated royalty �r(rp(�)) < �, and is smaller than the

expected benchmark royalty �� for a su¢ ciently weak patent.

24

This example illustrates the role of post-trial renegotiation in licensing an uncertain

patent. An individual challenge becomes less risky when it is possible to renegotiate ex

post a new royalty after the issue of the trial. Consequently, the patentee loses some of

its market power in determining ex ante the per-unit royalty rate that deters litigation

at equilibrium. For this reason refusing a license to an unsuccessful challenger should

not be allowed.

3 Fixed fee licensing schemes

In this section, licensing contracts o¤ered by the patentholder P to the n downstream

�rms involve a �xed fee only. The modelling leads to the same three-stage game as in

the per-unit royalty licensing scheme, simply replacing the royalty rate by a �xed fee

in the licensing contract o¤ered by the patentholder.

Denote �L(k) (respectively �NL(k)) the equilibrium pro�t of a downstream �rm pro-

ducing at a constant marginal cost c� � (respectively c) in an industry of n �rms, outof which k �rms produce at marginal cost c� � and the remaining n� k �rms produceat marginal cost c.

We introduce the following assumption which states that a licensee�s pro�t when all

�rms buy the license is higher than a non-licensee�s pro�t whatever the number of

licensees.

A8: �NL(k) < �L(n) for all k < n:

We start with a preliminary result describing what happens at equilibrium when not

all �rms accept the up-front fee.

Lemma 11 Consider a Nash equilibrium of stage 2. If not all �rms accept the licensingcontract in this equilibrium then there is at least one �rm (among those who do not

accept the contract) that challenges the patent validity.

Proof. Let us show that a situation where only k < n �rms accept the contract andnone of the remaining n � k �rms challenges the patent validity cannot be a Nashequilibrium of stage 2. If one of these �rms challenges the patent validity it gets an

expected pro�t of ��NL(k)+(1��)�L(n); whereas it gets a pro�t equal to �NL(k) if no�rm challenges the patent validity. From A8 it follows that ��NL(k) + (1� �)�L(n) >�NL(k) which means that a downstream �rm who does not accept the licensing contract

is always better o¤ challenging the patent validity.

25

In order to derive the demand function for licenses, we introduce the following assump-

tion:

A9: For all k between 0 and n� 1,

�L(k)� �L(k + 1) � �NL(k � 1)� �NL(k)

According to this assumption, a licensee�s incremental pro�t is at least equal to a non-

licensee�s incremental pro�t when the number of licensees is reduced by one unit. A

more precise interpretation of this assumption is given below.

3.1 Demand function for licenses: stage 2

The following proposition gives the demand for licenses at the Nash equilibrium of stage

2 as a function of the value of the up-front fee F chosen by the patentholder P in stage

1:

Proposition 12 Denote Fn(�) = ���L(n)� �NL(n� 1

�) and Fk = �L(k)��NL(k�1)

for all k � n� 1:If F � Fn(�) then the unique Nash equilibrium of stage 2 is the situation where all

downstream �rms accept the licensing contract.

If Fn(�) < F � Fn�1 then the Nash equilibria of stage 2 are the situations where

n� 1 downstream �rms accept the licensing contract and one �rm does not.

For any k between 0 and n� 2, if Fk+1 < F � Fk then the Nash equilibria of stage2 are the situations where k downstream �rms accept the licensing contract and the

remaining n� k �rms do not.If F > F1 then the unique Nash equilibrium of stage 2 is the situation where all

downstream �rms reject the licensing contract.

To avoid the multiple equilibria problem that arises when F is equal to one of the

threshold values Fk we assume that a downstream �rm which is indi¤erent between

accepting the license o¤er made by the patentholder and refusing it chooses to accept

it. Hence, we can de�ne the number k(F; �) of �rms that accept at equilibrium the

26

license o¤er F made by the patentholder:

k(F; �) =

8>>>>>>>>><>>>>>>>>>:

n if F � Fn (�)n� 1 if Fn (�) < F � Fn�1::: : :::

k if Fk+1 < F � Fk::: : :::

0 if F > F1

Note that k(F; �) depends on � only through the threshold Fn (�) : More precisely, if we

denote Fn(1) = Fn we have Fn(�) = �Fn and F > Fn (�) implies k(F; �) = k(F ):

3.2 Choice of the �xed fee: stage 1

The patentholder will choose F so as to maximize its licensing revenues anticipating the

number of downstream �rms that will accept the license o¤er. If the up-front fee F is

such that all �rms accept the o¤er then the patentholder�s licensing revenues are equal

to nF: If the up-front fee is such that there is at least one �rm that does not accept the

o¤er then litigation occurs and the patentholder gets licensing revenues equal to k(F )F

only when the patent validity is upheld by the court. This happens with probability �

which entails that the expected licensing revenues of the patentholder when F induces

a number of licensees k smaller than n are equal to �k(F )F: The expected licensing

revenues of the patentholder as a function of the up-front fee F can be summarized as

follows:

P (F; �) =

8>>>>>>>>><>>>>>>>>>:

nF if F � Fn (�)� (n� 1)F if Fn (�) < F � Fn�1

::: : :::

�kF if Fk+1 � F � Fk::: : :::

0 if F > F1

Since the demand function of licenses is stepwise, the maximization of P (F; �) with

respect to F will lead to one (or several) of the thresholds Fn (�) and F = Fk, k �n � 1: In other words, the maximization program max

F�0P (F; �) is equivalent to the

maximization program

maxF2fF1;:::;Fk;:::;Fn�1;Fn(�)g

P (F; �)

27

Since Fn (�) = �Fn, the expected licensing revenues P (F; �) for a value of F belonging

to the set fF1; :::; Fk; :::; Fn�1;Fn (�)g is given by:

P (F; �) =

8>>>>>>>>><>>>>>>>>>:

n(�Fn) if F = �Fn

� (n� 1)Fn�1 if F = Fn�1

::: : :::

�kFk if F = Fk

::: : :::

�F1 if F = F1

This shows that for any � 6= 0;maximizing P (F; �) over the set fF1; :::; Fk; :::; Fn�1; Fn (�)gis equivalent to maximizing P (F; 1) over the set fF1; :::; Fk; :::; Fn�1; Fng in the sensethat if the maximum of P (F; 1) is reached at Fk then the maximum of P (F; �) is reached

at Fk if k < n and at Fn (�) = �Fn if k = n: Hence, we have the following result:

Proposition 13 If the maximum of P (F; 1) is reached at F � = Fn then the paten-

tholder o¤ers a licensing contract with an up-front fee F � (�) = Fn(�) = �Fn that

induces a number of licensees equal to the total number of downstream �rms. If the

maximum of P (F; 1) is reached at F � = Fk with k < n then the patentholder o¤ers a

licensing contract with an up-front fee F � = Fk that induces a number of licensees equal

to k.

This proposition entails the following two results:

Corollary 14 The equilibrium number of licensees k� does not depend on the patent

strength �

Proof. The previous proposition shows that the choice of the �xed fee by the paten-tholder does not depend on �: Since the number of licensees is determined by the value

of the up-front fee �xed by the patentholder it follows that the equilibrium number of

licensees does not depend on the patent strength �:

Corollary 15 The equilibrium expected licensing revenues of the patentholder under

an up-front fee regime, denoted P �F (�) = P (F � (�) ; �), are proportional to the patent

strength, i.e.:

P �F (�) = �P�F (1)

28

Proof. If the patentholder o¤ers a licensing contract involving an up-front fee F =

Fn(�) = �Fn then its equilibrium licensing revenues are P �F (�) = n (�Fn) = � (nFn) =

�P �F (1): If the patentholder o¤ers a licensing contract involving an up-front fee F = Fkwhere k < n, then its equilibrium licensing revenues are P �F (�) = � (kFk) = �P

�F (1):

The results of this section lead to the conclusion that licensing an uncertain patent

by means of an up-front fee is not a¤ected by the uncertainty, in the sense that the

number of licensees does not depend on the patent strength and the patentholder�s

licensing revenues are exactly proportional to the patent strength. These results are

very di¤erent from those obtained with a per-unit royalty rate (previous section) or

with a two-part tari¤ as in Farrell and Shapiro (2007). This leads to a �rst conclusion:

licensing weak patents is very sensitive to the chosen licensing scheme. We must now

compare the licensing revenues collected through these schemes.

4 Royalty rate vs. �xed fee

In this section we show that, at least under some circumstances, the patentholder

prefers to use a royalty rate rather than an up-front fee in licensing contracts. Denote

P �r (�) = P (r� (�)) the optimal patentholder�s pro�t when the per-unit royalty licensing

scheme is used.

Proposition 16 If the patentholder gets higher licensing revenues when using the roy-alty rate scheme than with the �xed fee scheme when patent validity is perfect, i.e. � = 1;

it will also prefer to use a royalty rate rather than a �xed fee when the patent�s validity

is uncertain, i.e. � < 1:

Proof. This follows immediately from the fact that P �r (�) � �P �r (1) whereas P �F (�) =�P �F (1): Therefore, if P

�r (1) � P �F (1) then P

�r (�) � �P �r (1) � �P �F (1) = P �F (�) which

means that the patentholder�s licensing revenues are higher when the royalty rate mech-

anism is used.

This proposition gives only a su¢ cient condition for royalty rate contracts to be pre-

ferred over up-front fee contracts when the innovation is covered by an uncertain patent.

If royalties are preferred to �xed fees when � = 1; the former will be also preferred to

the latter when � < 1: However, this condition is far from necessary as the following

example shows: �xed fees may be preferred when � = 1 whereas royalties are preferred

for small values of �.

29

Example: Cournot competition with a linear demandWe know from Kamien and Tauman (1986) that with in a perfect patent setting, i.e.

� = 1; the patentholder�s licensing revenues are higher with an up-front fee than with

a royalty rate.14 We show hereafter that this ranking does not hold anymore when the

patent is uncertain: the patentholder may prefer to use the royalty rate mechanism

rather than the �xed fee mechanism.

We consider innovations of intermediate magnitude, i.e. a�cn�1 < � < a� c protected by

weak patents, i.e. � 2i0; �hwith � = 1 � [ n(a�c)

(n�1)(a�c+�) ]2. We do so because in this

case, we know the analytical expression of the royalty rate the patentholder will set,

i.e. r(�) = r2(�) = [1 �p1� �](a � c + �). This allows us to compute the quantity

produced at equilibrium by each downstream �rm: q (c� �+ r(�); n) =p1��(a�c+�)n+1

.

The equilibrium licensing revenues derived from the royalty r(�) = r2(�) are thus given

by:

P �r (�) = nr(�)q (c� �+ r(�); n) =n�p1� � � 1 + �

�(a� c+ �)2

n+ 1

Kamien and Tauman (1986, proposition 2) gives the patentholder�s pro�t expression

when � = 1. Using this expression and corollary (15), we derive the value of the

patentholder revenues for any � 2�a�cn�1 ; a� c

�:

P �F (�) =

(2�n

(n+1)2�2�a�c2�+ n+2

4

�2if a�c

n�1 < � �2(a�c)n

�n(n+2)

(n+1)2� (a� c) if 2(a�c)

n� � < a� c

Let us compare P �r (�) and P�F (�): First note that P

�F (�) is linear in � while P

�r (�)

is concave in �: Second, these functions take the same value for � = 0:We can then

state that a su¢ cient condition for P �r (�) to be greater than P�F (�) for all � 2

h0; �i

is that P �r (�) � P �F (�): The left-hand side of this inequality is given by P�r (�) =

nr(�)q�c� �+ r(�)

�= nrq (c� �+ r) = n2

(n�1)(n+1)��� a�c

n�1�(a� c) while the right-

hand side depends on whether � is such that a�cn�1 � � �

2(a�c)n

or 2(a�c)n

� � � a� c:Let us examine the subcase 2(a�c)

n� � � a � c: When this condition is satis�ed, we

have P �F (�) =n(n+2)

(n+1)2� (a� c)

�1� [ n(a�c)

(n�1)(a�c+�) ]2�: Comparing P �r (�) to P

�F (�) amounts

then to compare nn�1

��� a�c

n�1�to n+2

n+1��1� [ n(a�c)

(n�1)(a�c+�) ]2�:A su¢ cient (and necessary)

condition to have P �r (�) � P �F (�) is�

nn�1 �

n+2n+1

��� n

n�1a�cn�1 � �

n+2n+1�[ n(a�c)(n�1)(a�c+�) ]

2: The

14Sen (2005) shows that this result holds only when the number of �rms in the downstream industryis not too high.

30

left-hand side of this inequality is clearly increasing in � while it is straighforward to

show that the right-hand side is decreasing in �: Therefore , to show that the previous

inequality holds for any � 2h2(a�c)n; a� c

i; it is su¢ cient to show that it holds for

� = 2(a�c)n. Taking the inequality for � = 2(a�c)

nand simplifying by (a� c), we get

nn�1

�2n� 1

n�1�> 2(n+2)

n(n+1)

�1� n

(n�1)(1+ 2n)

�2which can be shown after some algebraic

manipulations to be equivalent to nn�1 >

2n+1, which is obviously true. Thus, the

condition P �r (�) � P �F (�) holds for an innovation such that2(a�c)n

� � � a� c: We canthen state the following result:

Proposition 17 If the downstream �rms compete à la Cournot in a market where the

demand is linear, then for an innovation � of intermediate magnitude, i.e. such that2(a�c)n

� � � a � c; covered by a relatively weak patent, i.e. such that � � �, the

patentholder gets higher licensing revenues using a royalty rate rather than an up-front

fee, whereas if the patent were perfect the inverse would be true.

It is important to note that the per-unit royalty licensing scheme is preferred in this

probabilistic right framework only because the patent is uncertain, while a possible

preference for this licensing scheme in the framework of a perfect protection rests on

a completely di¤erent reason, mainly related to the size of the downstream industry

(Sen, 2005).

5 Conclusion

The consequences of licensing uncertain patents have been examined in this paper by

addressing the following question: to what extent licensing a patent that has a positive

probability to be invalidated if it is challenged favors the patentholder when confronted

to potential users in an oligopolistic downward industry? Our results show that the

answer to Farrell and Shapiro�s question "How strong are weak patents?" is very sensi-

tive to the choice of the licensing scheme. Two licensing schemes have been examined:

the per-unit royalty rate and the up-front fee. The most salient result is that these two

mechanisms lead to opposite consequences. While licensing uncertain patents by means

of a royalty rate allows in general the patentholder to reap some extra pro�t relative to

the expected pro�t after the court resolution of the patent validity, a �xed fee regime

discards completely this possibility. Under a �xed fee the patentholder obtains exactly

31

its expected revenue. These results mainly arise from letting the number of licensees

depend on the price of the license chosen by the patentholder, either a per-unit royalty

rate or an up-front fee. The second important result is that under the per-unit royalty

licensing regime the holder of a weak patent may prefer to sell at the same royalty rate

as if the patent was certain, taking thus the risk of triggering a litigation on patent

validity. However the justi�cation of this result is completely di¤erent from Farrell and

Shapiro (2007). It is precisely when the royalty rate acceptable by all the �rms in the

downward industry is too low that the holder of a weak patent prefers to sell at the

royalty rate that maximizes its licensing revenues as if the patent was certain. Moreover

we have shown that even if �xed fees are preferred when the patent is very strong, roy-

alties may be more pro�table if the patent is uncertain, particularly if it is weak. The

classical properties of licensing certain patents may thus be reversed in the uncertain

patent framework. We have also explored di¤erent policy levers a¤ecting the paten-

tholder�s market power when using a per-unit royalty rate. We showed that its market

power may be reduced in two ways: First, by preventing the patentholder�s refusal to

sell a license to an unsuccessful challenger. Second, by favoring collective challenges of

patents�validity, particularly when competition intensity in the downstream market is

so high that individual incentives to challenge a patent are weak.

One important question concerns the patent quality problem. Since the patent

system involves a two-tier process combining patent o¢ ce examination and challenge

by a court of the validity of the granted patent, there are two possible approaches to

this problem.

The �rst approach is to �nd some ways to encourage third parties to bring to a

court pieces of evidence in order to challenge the validity of presumably weak patents

(post-grant opposition in Europe or post-grant reexamination in the United States).

Giving more incentives to potential licensees to challenge a patent validity is necessary

insofar as the free riding aspect weakens individual incentives. In this perspective, two

policy levers are suggested by our model: the renegotiation of the licensing contract

with an unsuccessful challenger and the cooperative approach among potential licensees

to collectively accept or refuse a licensing contract. Incentives to renegotiate could be

encouraged by not allowing a patentee to refuse a license to an unsuccessful chalenger.

Allowing a joint decision for accepting or refusing a licensing contract may also reduce

the patentholder�s market power.

The second approach to the patent quality problem is to improve the screening

32

process inside the patent o¢ ce itself through the strengthening of the patentability

standards, turning back the Lemley�s "rational ignorant patent o¢ ce principle" (Lem-

ley, 2001). This second approach could be interesting, particularly when the patent

strength is no more common knowledge but a private information parameter (Chiou,

2008). The patent o¢ ce could thus propose to any applicant a menu involving the

choice of either paying an extra fee to obtain a thorough examination process at the

patent o¢ ce signalling thus a high patent quality or paying a lower fee to simply obtain

a "standard" examination process that may signal the weakness of the patent. Design-

ing an e¢ cient mechanism to implement such a procedure is left for future investigation.

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APPENDIX

Proof of Lemma 1Condition (1) is equivalent to:

�(c� �+ r; c� �+ r)� ��(c; c� �+ r)� (1� �)�(c� �; c� �) � 0

A3 implies that �(c � � + r; c � � + r) is strictly decreasing in r, and A2 implies that�(c; c��+r) is increasing in r. It follows that g(r) = �(c��+r; c��+r)���(c; c��+r)�(1��)�(c��; c��) is strictly decreasing in r and continuous (by A1). Furthermore,g(0) = �(c��; c��)���(c; c��)�(1��)�(c��; c��) = � (�(c� �; c� �)� �(c; c� �)) �0 and g (�) = �(c; c)���(c; c)�(1��)�(c��; c��) = (1� �) (�(c; c)� �(c� �; c� �)) �0: Therefore, there exists a unique solution to the equation g(r) = 0 and this solution,

denoted r(�); belongs to the interval [0; �] : Moreover, since g is strictly decreasing, the

condition g(r) � 0 is equivalent to r � r (�) :

35

Proof of Proposition 2Di¤erentiating the equation � (c� �+ rc(�); c� �+ rc(�)) = �� (c; c)+(1� �)� (c� �; c� �)with respect to �, we get dr

c(�)d�

= �(c;c)��(c��;c��)(�1+�2)(c��+rc(�);c��+rc(�)) . Both the numerator and the

denominator are negative which implies that rc(�) is increasing.

(i) Since � (c; c)� � (c� �; c� �) < 0 (A4), drc(�)d�

is decreasing in � over [0; 1] (i.e. rc(�)

is concave) if and only if (�1 + �2) (c� �+ rc(�); c� �+ rc(�)) is decreasing in � over[0; 1] : Since rc(�) is continuous and strictly increasing from rc(0) = 0 to rc(1) = �;

the latter condition is equivalent to (�1 + �2) (x; x) is decreasing in x over [c� �; c] ;which means that x ! �(x; x) is concave over [c� �; c]. In this case, rc(�) � �rc(1) +(1� �) rc(0) = ��:(ii) can be shown in a similar way.

Proof of Proposition 4We �rst show that if r < � it is impossible to have an equilibrium in which the number

k of �rms accepting the o¤er is strictly less than n� 1. If this was true then one of then�k � 2 �rms that have not accepted the licensing contract could get a higher expectedpro�t by deviating unilaterally and accepting the contract. Indeed, if it deviates then

litigation will still occur because there will remain at least one �rm refusing the license

o¤er. This would result in the deviating �rm having a marginal cost c� � + r insteadof c in case the patent is upheld, while still having a marginal cost equal to c� � if thepatent is invalidated by the court. Hence, the number of �rms accepting the license

o¤er r < � at equilibrium is at least equal to n�1: This remains true for r = � under theassumption that a �rm accepts the o¤er when indi¤erent between accepting or refusing

it. Furthermore, if r � r(�), condition (1) shows that an equilibrium cannot involve

k = n� 1 licensees: Thus i/ is proven.If r > r(�); an outcome in which one �rm refuses the license o¤er while the others

accept it is a Nash equilibrium: condition (1) shows that the �rm refusing the o¤er gets

a higher pro�t than if it had accepted it, and it has been shown that the remaining �rms

do not bene�t from refusing the license since the patent will be challenged anyway: This

proves ii/.

Part iii/ of the proposition is straightforward.

Proof of Lemma 5Let k 2 fn� 1; ng : Since the function rq (c� "+ r; k � 1) is concave in r and reachesits maximum at ~rk (�) then it is increasing over [0; ~rk (�)] : Consequently, the following

36

holds:

� � ~rk (�)()@

@r(rq (c� "+ r; k)) jr=� � 0() q (c; k)+ �

@

@r(q (c� "+ r; k)) jr=� � 0

Let us compare @@r(q (c� "+ r; n)) jr=� and @

@r(q (c� "+ r; n� 1)) jr=�:It is clear that

q(c; n) = q(c; n � 1): both expressions refer to the individual output of a �rm in a

symmetric oligopoly consisting of n �rms producing at marginal cost c: Thus, using as-

sumption A5, we get: q(c�"+r;n�1)�q(c;n�1)r�� � q(c�"+r;n)�q(c;n)

r�� for all r < �: Taking the limit

of both sides as r ! �, we obtain @@r(q (c� "+ r; n)) jr=� � @

@r(q (c� "+ r; n� 1)) jr=�.

Hence, q (c; n) + � @@r(q (c� "+ r; n)) jr=� � q (c; n� 1) + � @

@r(q (c� "+ r; n� 1)) jr=�:

Therefore, the following chain of implications holds:

� � ~rn�1 (�) =) q (c; n� 1) + � @@r(q (c� "+ r; n� 1)) jr=� � 0

=) q (c; n) + �@

@r(q (c� "+ r; n)) jr=� � 0 =) � � ~rn (�)

Proof of Proposition 6Assume that � � ~rn�1 (�) : By lemma (5), the inequality � � ~rn (�) holds as well. In

this case the maximum of P �(r) over [0; r(�)] is reached at r(�); and its maximum

over ]r(�); �] is reached at �: Therefore, we must compare nr(�)q(c � � + r(�); n) to(n� 1)��q(c; n� 1): Consider a royalty rate r 2 [0; �] : The inequality nrq(r; n) � (n�1)��q(�; n) is ful�lled if and only if rq(c��+r;n)

�q(c;n)� n�1

n�: Since the function r �! rq(c��+r;n)

�q(c;n)

is strictly increasing and continuous in r and takes the value 0 for r = 0 and 1 for r = �,

there exists a unique solution to the equation rq(c��+r;n)�q(c;n)

= n�1n�, which is denoted

by s (�) : The condition rq(c��+r;n)�q(c;n)

� n�1n� can then be written as r � s (�) : Hence

the inequality nr(�)q(c � � + r(�); n) � n � 1)��q(c; n) amounts to r (�) � s (�) : Theconvexity of s (�) can be derived from the concavity of w : r �! rq(c � � + r; n) andits increasingness over [0; �] : di¤erentiating twice the equation w(s(�))

w(�)= �n�1

n, we get

w00 (s (�)) (s0 (�))2+w0 (s (�)) s00 (�) = 0 which leads to s00 (�) = �w00(s(�))(s0(�))2

w0(s(�)) > 0: The

property s(0) = 0 is immediate and the property s(1) < � derives from n�1n< 1:

Proof of Proposition 7Assume that ~rn�1 (�) < � � ~rn (�) : In this case, the maximum of P (r) over [0; r(�)] is

reached at r (�) : De�ne ~�n�1 as the unique solution in � to the equation r (�) = ~rn�1 (�)

37

(it is straightforward to check that such a solution exists in [0; 1] and is unique):Two

subcases must be distinguished:

- Subcase 1: � � ~�n�1 : The maximum of P (r) over ]r(�); "] is then reached at ~rn�1 (�) :Determining the royalty rate that maximizes P (r) over [0; �] amounts then to the com-

parison of nr(�)q(c��+r(�); n) and (n�1)�~rn�1 (�) q(c��+~rn�1 (�) ; n�1): The formeris greater than the latter if and only if r (�) is greater than v (�) de�ned as the unique

solution in r to the equation nrq(c� �+r; n) = (n�1)�~rn�1 (�) q(c� �+~rn�1 (�) ; n�1).The existence, uniquess, increasingness and convexity with respect to � of such a so-

lution can be established in a similar way to that of s (�) : The function v (�) satis�es

as well the properties v (0) = 0 and v (1) < �: The �rst inequality is straightforward

to show and the second one derives from n"q(c; n) > n~rn�1 (�) q(c � � + ~rn�1 (�) ; n)(which holds because ~rn�1 (�) < � � ~rn (�)) and n~rn�1 (�) q(c � � + ~rn�1 (�) ; n) >

(n� 1) ~rn�1 (�) q(c��+~rn�1 (�) ; n�1): Indeed these two inequalities result in n"q(c; n) >(n� 1) ~rn�1 (�) q(c � � + ~rn�1 (�) ; n � 1) = nv (1) q(c � � + v (1) ; n) and consequentlylead to � > v (1) :

- Subcase 2: � > ~�n�1 : The upper bound of P (r) over ]r(�); "] is then reached at r(�)+:

From the expression of P (r), it is clear that P (r (�)) > P (r (�)+): Hence, the maximum

of P (r) over [0; �] is reached at r(�):Consequently litigation is always deterred in this

subcase.

Proof of Proposition 8Assume that � > ~rn (�) : By lemma (6) the inequality � > ~rn�1 (�) holds as well. Analo-

gously to ~�n�1;de�ne ~�n as the unique solution in � to the equation r (�) = ~rn (�). Three

subcases are distinguished:

- Subcase 1: � � min�~�n�1; ~�n

�: The maximum of P (r) over [0; r(�)] is then reached at

r (�) and its maximum over ]r(�); "] is reached at ~rn�1 (�) : Hence the analysis conducted

in subcase 1 in the proof of proposition 7 applies here.

- Subcase 2 : ~�n�1 < � < ~�n : The maximum of P (r) over [0; r(�)] is then reached at

r (�) and its maximum over ]r(�); "] is reached at r(�)+: Therefore the maximum of

P (r) over [0; �] is reached at r(�) (see subcase 2 in the proof of proposition 7) which

implies that litigation is deterred. Note that this subcase is not relevant if the inequality~�n�1 < ~�n does not hold.

- Subcase 3: � � ~�n : The maximum of P (r) over [0; r(�)] is then reached at ~rn (�) :

This is su¢ cient to state that the maximum of P (r) over [0; �] is reached at ~rn (�) :This

follows from the fact that the function r �! nrq(c� �+ r; n) reaches its unconstrained

38

maximum at ~rn (�) and nrq(c� �+ r; n) > �(n� 1)rq(c� �+ r; n� 1) for any r 2 [0; �] :The latter inequality results from assumption A7: nq(c� �+ r; n) = Q(c� �+ r; n) �Q(c� �+ r; n� 1) � (n� 1)q(c� �+ r; n� 1):

Proof of Corollary 9Assume that r (�) > ��: Since s(�) is a convex function such that s (0) = 1 and s (1) < �

then s(�) � �� for all � 2 [0; 1]. Consequently a su¢ cient condition for the inequalityr (�) � s (�) to hold is that r (�) � ��: The same conclusion applies for the convex

function v(�): Given this, the �rst part of the corollary follows immediately from the

three previous propositions.

Using the three previous propositions, it is straighforward to check that under the

conditions r (�) � s (�) and r (�) � v (�) (which hold when r (�) > ��), the optimal

royalty rate set by the patentholder simpli�es as follows: r� (�) = min (r (�) ; ~rn (�)) :

Using the inequality r (�) > ��, we get r� (�) > min (��; ~rn (�)) > min (��; �~rn (�)) =

�min (�; ~rn (�)) = �r� (1) : Hence for all � 2 [0; 1] ; �r� (1) < r� (�) = min (r (�) ; ~rn (�)) :

Since the function P (r) = nrq(c��+r; n) is concave in r over [0; r (�)] then P (�r� (1)) >�P (r� (1)) and since it reaches its maximum at ~rn (�), it is increasing over [0; r� (�)] which

entails that P (r� (�)) > P (�r� (1)) : From the two previous inequalities, we obtain that

P (r� (�)) > �P (r� (1)):

Proof of Proposition 12The situation where the n �rms accept the licensing contract F is a Nash equilibrium

if and only if:

�L(n)� F � ��NL(n� 1) + (1� �)�L(n)

which can be rewritten as:

F � ���L(n)� �NL(n� 1

�) (6)

that is

F � Fn(�)

A situation where n� 1 �rms accept the licensing contract and one �rm does not is a

Nash equilibrium (of stage 2) if and only if:

��NL(n� 1) + (1� �)�L(n) � �L(n)� F (7)

39

and

�[�L(n� 1)� F ] + (1� �)�L(n) � ��NL(n� 2) + (1� �)�L(n) (8)

Condition (7) means that the one �rm that does not accept the licensing contract

and challenges the patent�s validity does not �nd it optimal to unilaterally deviate by

accepting the licensing contract. Condition (8) means that none of the n � 1 �rmswhich accept the licensing contract �nd it optimal to unilaterally deviate by refusing

the contract. When the number of �rms accepting the contract is strictly less than n,

litigation will occur (lemma 11) which entails that the �rms accepting the contract pay

the �xed fee F only if the patent validity is upheld, which happens with probability �:

With the complementary probability 1 � �; the patent is invalidated and all the �rmsget the same pro�t namely �L(n): It is straightforward to show that conditions (7) and

(8) are equivalent to the following double inequality:

�[�L(n)� �NL(n� 1)] � F � �L(n� 1)� �NL(n� 2)

that is

Fn(�) � F � Fn�1

Note that the inequality �[�L(n) � �NL(n � 1)] < �L(n � 1) � �NL(n � 2) followsimmediately from A9 for � = 1 and is a fortiori satis�ed for � < 1:

A situation where k � n � 2 �rms accept the licensing contract and the remaining donot is a Nash equilibrium of the stage 2 subgame if and only if:

���L(k)� F

�+ (1� �)�L(n) � ��NL(k � 1) + (1� �)�L(n) (9)

and

��NL(k) + (1� �)�L(n) � ���L(k + 1)� F

�+ (1� �)�L(n) (10)

Condition (9) means that none of the k �rms accepting the licensing contract �nds it

optimal to unilaterally deviate by refusing the contract and condition (10) means that

none of the n � k �rms refusing the licensing contract �nds it optimal to unilaterallydeviate by accepting the contract. It is easy to see that conditions (9) and (10) can be

combined into the following double inequality that does not depend on �:

�L(k + 1)� �NL(k) � F � �L(k)� �NL(k � 1)

40

that is:

Fk+1 � F � Fk

Note that the inequality �L(k + 1) � �NL(k) � �L(k) � �NL(k � 1) follows from A9.

Thus, the role of assumption A9 is to guarantee that the set of values of F belonging

to the interval [Fk+1; Fk] is not empty.

A situation where no �rm accepts the licensing contract is a Nash equilibrium if and

only if:

��NL(0) + (1� �)�L(n) � ���L(1)� F

�+ (1� �)�L(n)

which can be rewritten as:

�NL(0) � �L(1)� F

or equivalently as:

F � �L(1)� �NL(0) = F1

41